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Full text of "Mathematical And Physical Papers - Iii"

58     ON THE EFFECT  OF  THE INTERNAL  FRICTION OF FLUIDS
general equations have the same form whether the fluid be referred to the fixed or moveable axes. But on the latter supposition the equations of condition (30) become rigorously exact. Hence equation (40) gives correctly the solution of the problem, independently of the restriction that the maximum excursion of the sphere be small compared with its radius, provided we suppose the polar co-ordinates r, 6 measured from the centre of the sphere in its actual, not its mean position. Similar remarks apply to the problem of the cylinder. Moreover, in the case of a sphere oscillating within a concentric spherical envelope, it is not necessary, in order to employ the solution obtained in Section II., that the maximum excursion of the sphere be small compared with its radius ; it is sufficient that it be small compared with the radius of the envelope.
These are points of great importance, because the excursions of an oscillating sphere in a pendulum experiment are not by any means extremely small compared with the radius of the sphere ; and in the case of a narrow cylinder, such as the suspending wire, so far from the maximum excursion being small compared with the radius of the cylinder, it is, on the contrary, the radius which is small compared with the maximum excursion.
42. Let us now return to the case of the uniform motion of a sphere. In order to obtain directly the expression for the resistance of the fluid, it would be requisite first to find p, then to get Pr and T0 from (46), or at least to get the values of these functions for r = a, and lastly to substitute in (47) and perform the integration. We should obtain p by integrating the expression for dp got from (16) and (17). It would be requisite first to express u and q in terms of -^, then to transform the expression for dp so as to involve polar co-ordinates, and then substitute for -^ its value given by (121) ; or else to express the right-hand member of (121) by the co-ordinates #, vr, and substitute in the expression for cúp*. We
* The equations (16), (17) give, after a troublesome transformation to polar co-
ordinates,
dp_     fjt,      d^ / d?     sin0 d^    I    d     p d\
3r~VsiiT<? dB \dr* + ~1*~ d6 sm0 dd ~ ^di) * .................. W*
d$_ __ /*_ ^ /_tf2    ging d_   1    d_    p  d\
1e~~~ sO dr vP + ~r2~ d6 s5f0 50 ~ ^ Jt) ^ .................. ( '*
The expression for dp got from these equations is an exact differential by virtue of the equation which determines \j/ Ľ and in the problems considered in Section IL